Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Ten-Face Non-Edge-Sharing Wing Set on the Regular Icosahedron and a Decagonal Equatorial Balance

YoungJune Jeon

Abstract

We formalize a ten-face triangular wing set on a regular icosahedron under a vertex labeling N, S, U1-U5, L1-L5 with rotation axis NS. The wing faces satisfy: (i) each face is an isosceles 36-36-108 triangle with a 36-degree angle anchored at a pole (N or S); (ii) distinct faces may share vertices but share no edges; and (iii) a natural equatorial cross-section yields a perfectly balanced regular decagon. We derive a closed form for the decagon radius, R = (phi/2)*ell, where ell is the icosahedron edge length and phi is the golden ratio phi = (1 + sqrt(5))/2. Beyond the geometric results, we interpret the ten-face closure as a symmetry-consistent design principle for a pole-anchored wing layout and provide a reproducible construction workflow.

A Ten-Face Non-Edge-Sharing Wing Set on the Regular Icosahedron and a Decagonal Equatorial Balance

Abstract

We formalize a ten-face triangular wing set on a regular icosahedron under a vertex labeling N, S, U1-U5, L1-L5 with rotation axis NS. The wing faces satisfy: (i) each face is an isosceles 36-36-108 triangle with a 36-degree angle anchored at a pole (N or S); (ii) distinct faces may share vertices but share no edges; and (iii) a natural equatorial cross-section yields a perfectly balanced regular decagon. We derive a closed form for the decagon radius, R = (phi/2)*ell, where ell is the icosahedron edge length and phi is the golden ratio phi = (1 + sqrt(5))/2. Beyond the geometric results, we interpret the ten-face closure as a symmetry-consistent design principle for a pole-anchored wing layout and provide a reproducible construction workflow.
Paper Structure (13 sections, 10 theorems, 30 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A standard coordinate model
  4. Labeling and Wing-Face Axioms
  5. Figures
  6. Main Results
  7. Angle structure via a golden-distance lemma
  8. Edge non-sharing and maximality
  9. Equatorial decagon and closed-form radius
  10. Design Interpretation: Why "Ten"?
  11. Reproducible Construction Workflow (CAD/Parametric)
  12. Related Geometric Context
  13. Conclusion

Key Result

Lemma 1

In the standard coordinate model eq:std-icosa (edge length $2$), (i) adjacent vertices have distance $2$, and (ii) for an appropriate choice of opposite poles $N,S$ and a corresponding upper vertex $U$, the distance between a pole and an adjacent upper vertex is $2$, whereas the distance between the

Figures (3)

  • Figure 1: Labeled regular icosahedron used throughout the paper. The rotation axis is the line $NS$ (aligned with the $z$-axis). The upper-ring vertices are labeled $U_1,\dots,U_5$ and the lower-ring vertices $L_1,\dots,L_5$.
  • Figure 2: The GeoWind ten-face wing set $\mathcal{F}$ embedded in the labeled icosahedron. South-anchored faces are $F_S(i)=\triangle(S,U_i,L_i)$ and north-anchored faces are $F_N(i)=\triangle(N,U_i,L_{i-1})$ (indices modulo 5). Faces are shown translucent; vertex sharing is allowed, but no two faces share an edge.
  • Figure 3: Equatorial plane $\Pi$ (perpendicular to axis $NS$ through center $O$). The ten representative points $p(F)$ (midpoints of the pole-opposite edges) lie on a circle centered at $O$ and form a regular decagon of radius $R=\frac{\varphi}{2}\ell$.

Theorems & Definitions (22)

  • Definition 1: Vertex labeling
  • Definition 2: GeoWind ten-face wing set
  • Lemma 1: Edge and diagonal distances in the standard model
  • proof
  • Lemma 2: Cosine-law characterization of the golden gnomon
  • proof
  • Theorem 1: Golden-triangle shape: $36^\circ\!-\!36^\circ\!-\!108^\circ$
  • proof
  • Theorem 2: No shared edges within the ten faces
  • proof
  • ...and 12 more